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  "body": "\n## Summary\n\nA square-integrable kernel is a [[kernel|GSMBXT7FPK]] $K(x, s)$ whose squared absolute value has a finite double integral.\nSuch kernels are also called Hilbert–Schmidt kernels or $L^2$ kernels.\nThe integral operator of a square-integrable kernel is a bounded and [[compact operator|M2NEP1K5AE]] on the [[Lp space|SASP14MA1Z]] $L^2$, namely a [[Hilbert–Schmidt integral operator|CB5G212Q03]].\nThis class contains all continuous kernels on a square and also many unbounded ones.\n\n## Explanations\n\n### L1 human\n\nThe kernel may be infinite at some points, but not too badly: its square must still have a finite total integral.\nThis is the natural condition when one measures functions by their energy $\\int \\abs{\\varphi}^2$.\nFor example, $K(x, s) = \\abs{x - s}^{-1/4}$ on $[0, 1]^2$ is unbounded but square-integrable.\nThe kernel $\\abs{x - s}^{-1/2}$ is not, because the double integral of $1/\\abs{x - s}$ diverges.\n",
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